Question

If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log  3/2 - 1/3))` is equal to ______.

Options

• 3

• 4

• 1

• 2

Answer 1

If y = `(1 + 1/x)^x` then `(2sqrt(y_2(2) + 1/8))/((log  3/2 - 1/3))` is equal to 3.

Explanation:

Let y = `(1 + 1/x)^x`

Taking logarithm of both sides, we get

log y = `x[log (1 + 1/x)]`

`\implies 1/y y_1(x) = x^2/(x + 1)(-1/x^2) + log(1 + 1/x)`

= `-1/(x + 1) + log(1 + 1/x)`  .........(i)

Since, y (2) = `(1 + 1/2)^2 = 9/4`

So, y₁ (2) = `(9/4)(-1/3 + log  3/2)`

Again differentiate equation (i) w.r.t (x), we get

`(y(x)y_2(x) - [y_1(x)]^2)/(y(x))^2 = 1/(1 + x)^2 - 1/(x(x + 1))`

By putting x = 2, we get

`(y(2)y_2(2) - (y_1(2))^2)/(y(2))^2 = (-1)/8`

Now, put value of y(2) and y₁(2)

`\implies` y₂ (2) = `(9/4)(-1/3 + log  3/2)^2 - 1/8`

`(y_2(2) + 1/8)^4 = 9(log  3/2 - 1/3)^2`

`\implies` Required expression = 3